The catenoid estimate and its geometric applications

TL;DR

This paper introduces a sharp area estimate for catenoids that advances min-max theory applications, including minimal surface existence and properties in positively curved manifolds, with implications in higher dimensions.

Contribution

It provides a new sharp area estimate for catenoids and applies it to solve several longstanding problems in minimal surface theory and geometric analysis.

Findings

Width of three-manifolds with positive Ricci curvature is realized by an orientable minimal surface.

Minimal genus Heegaard surfaces can be isotoped to minimal surfaces in such manifolds.

Constructs variationally the doublings of Clifford torus via equivariant min-max.

Abstract

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index 1 orientable minimal hypersurface.